Hi, I've found a proof from $\displaystyle |x| \leq \sum_{i=1}^{n} |x_{i}|$ given $\displaystyle x=(x_{1},...,x_{n}), n \in \mathbb{N}$ It reads:

$\displaystyle \sum_{i=1}^{n}(x_{i})^{2} \leq \sum_{i=1}^{n} (x_{i})^2 + 2\sum_{i \ne j}^{n} |x_{i}||x_{j}| = (\sum_{i=1}^{n}|x_{i}|)^2$.

And taking the square root it's supposed to grant the result.

Now, I do not see why $\displaystyle a^2 \geq b^2 \Rightarrow a \geq b$

Thanks in advance